In a pie plate or a shallow dish, add the flour, 1 tablespoon of the salt, and the black pepper and whisk to combine.
Add the chicken breasts to the seasoned flour and turn them to coat fully on both sides.
Shake off any excess flour and set the chicken aside.
In a large, heavy-bottomed skillet over medium-high heat, add 3 tablespoons of the vegetable oil.
Add the chicken breasts to the hot oil and cook, undisturbed, until the bottoms are a deeply golden color, about 3 minutes.
Flip the chicken breasts and cook, undisturbed, until the reverse sides are deeply golden-brown, about 3 minutes.
Transfer the chicken breasts to a plate.
Add the remaining vegetable oil to the same skillet over medium-high heat and allow it to heat.
Add the mushrooms to the hot oil and cook, while stirring occasionally, until they begin to brown and release their liquid, about 5 minutes.
Add the garlic and cook, while stirring, until it is fragrant, about 30 seconds.
Add the cream cheese to the mushroom-garlic mixture and cook, while stirring, until it is melted.
Add the half-and-half and stir to combine.
Using a wooden spoon, scrape the bottom of the skillet to release the browned bits.
Season the mushroom mixture with the extra salt and the black pepper.
Add the chicken breasts to the skillet and cover the skillet.
Cook, turning the chicken breasts halfway through, until they reach an internal temperature of 165 degrees F, about 15-20 minutes.
Transfer the chicken breasts to a plate and tent them with foil to keep them warm.
Add the spinach to the mushroom mixture and cover the skillet.
Cook until the spinach is wilted, about 5 minutes.
Stir the spinach mixture.
Season the spinach mixture with the extra salt and the black pepper.
Serve the chicken breasts with the spinach mixture over the egg noodles.